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  <section id="kane-s-method-in-physics-mechanics">
<span id="kane-method"></span><h1>Kane’s Method in Physics/Mechanics<a class="headerlink" href="#kane-s-method-in-physics-mechanics" title="Permalink to this headline">¶</a></h1>
<p><a class="reference internal" href="index.html#module-sympy.physics.mechanics" title="sympy.physics.mechanics"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sympy.physics.mechanics</span></code></a> provides functionality for deriving equations of motion
using Kane’s method <a class="reference internal" href="reference.html#kane1985" id="id1"><span>[Kane1985]</span></a>. This document will describe Kane’s method
as used in this module, but not how the equations are actually derived.</p>
<section id="structure-of-equations">
<h2>Structure of Equations<a class="headerlink" href="#structure-of-equations" title="Permalink to this headline">¶</a></h2>
<p>In <a class="reference internal" href="index.html#module-sympy.physics.mechanics" title="sympy.physics.mechanics"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sympy.physics.mechanics</span></code></a> we are assuming there are 5 basic sets of equations needed
to describe a system. They are: holonomic constraints, non-holonomic
constraints, kinematic differential equations, dynamic equations, and
differentiated non-holonomic equations.</p>
<div class="math notranslate nohighlight">
\[\begin{split}\mathbf{f_h}(q, t) &amp;= 0\\
\mathbf{k_{nh}}(q, t) u + \mathbf{f_{nh}}(q, t) &amp;= 0\\
\mathbf{k_{k\dot{q}}}(q, t) \dot{q} + \mathbf{k_{ku}}(q, t) u +
\mathbf{f_k}(q, t) &amp;= 0\\
\mathbf{k_d}(q, t) \dot{u} + \mathbf{f_d}(q, \dot{q}, u, t) &amp;= 0\\
\mathbf{k_{dnh}}(q, t) \dot{u} + \mathbf{f_{dnh}}(q, \dot{q}, u, t) &amp;= 0\\\end{split}\]</div>
<p>In <a class="reference internal" href="index.html#module-sympy.physics.mechanics" title="sympy.physics.mechanics"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sympy.physics.mechanics</span></code></a> holonomic constraints are only used for the linearization
process; it is assumed that they will be too complicated to solve for the
dependent coordinate(s).  If you are able to easily solve a holonomic
constraint, you should consider redefining your problem in terms of a smaller
set of coordinates. Alternatively, the time-differentiated holonomic
constraints can be supplied.</p>
<p>Kane’s method forms two expressions, <span class="math notranslate nohighlight">\(F_r\)</span> and <span class="math notranslate nohighlight">\(F_r^*\)</span>, whose sum
is zero. In this module, these expressions are rearranged into the following
form:</p>
<blockquote>
<div><p><span class="math notranslate nohighlight">\(\mathbf{M}(q, t) \dot{u} = \mathbf{f}(q, \dot{q}, u, t)\)</span></p>
</div></blockquote>
<p>For a non-holonomic system with <span class="math notranslate nohighlight">\(o\)</span> total speeds and <span class="math notranslate nohighlight">\(m\)</span> motion constraints, we
will get o - m equations. The mass-matrix/forcing equations are then augmented
in the following fashion:</p>
<div class="math notranslate nohighlight">
\[\begin{split}\mathbf{M}(q, t) &amp;= \begin{bmatrix} \mathbf{k_d}(q, t) \\
\mathbf{k_{dnh}}(q, t) \end{bmatrix}\\
\mathbf{_{(forcing)}}(q, \dot{q}, u, t) &amp;= \begin{bmatrix}
- \mathbf{f_d}(q, \dot{q}, u, t) \\ - \mathbf{f_{dnh}}(q, \dot{q}, u, t)
\end{bmatrix}\\\end{split}\]</div>
</section>
<section id="id2">
<h2>Kane’s Method in Physics/Mechanics<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h2>
<p>The formulation of the equations of motion in <a class="reference internal" href="index.html#module-sympy.physics.mechanics" title="sympy.physics.mechanics"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sympy.physics.mechanics</span></code></a> starts with
creation of a <code class="docutils literal notranslate"><span class="pre">KanesMethod</span></code> object. Upon initialization of the
<code class="docutils literal notranslate"><span class="pre">KanesMethod</span></code> object, an inertial reference frame needs to be supplied. along
with some basic system information, such as coordinates and speeds</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s1">&#39;N&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">u1</span><span class="p">,</span> <span class="n">u2</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s1">&#39;q1 q2 u1 u2&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span><span class="p">,</span> <span class="n">u1d</span><span class="p">,</span> <span class="n">u2d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s1">&#39;q1 q2 u1 u2&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">],</span> <span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">])</span>
</pre></div>
</div>
<p>It is also important to supply the order of coordinates and speeds properly if
there are dependent coordinates and speeds. They must be supplied after
independent coordinates and speeds or as a keyword argument; this is shown
later.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s1">&#39;q1 q2 q3 q4&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">,</span> <span class="n">u4</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s1">&#39;u1 u2 u3 u4&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Here we will assume q2 is dependent, and u2 and u3 are dependent</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># We need the constraint equations to enter them though</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span><span class="p">],</span> <span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u4</span><span class="p">])</span>
</pre></div>
</div>
<p>Additionally, if there are auxiliary speeds, they need to be identified here.
See the examples for more information on this. In this example u4 is the
auxiliary speed.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span><span class="p">],</span> <span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">],</span> <span class="n">u_auxiliary</span><span class="o">=</span><span class="p">[</span><span class="n">u4</span><span class="p">])</span>
</pre></div>
</div>
<p>Kinematic differential equations must also be supplied; there are to be
provided as a list of expressions which are each equal to zero. A trivial
example follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">kd</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">u1</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">u2</span><span class="p">]</span>
</pre></div>
</div>
<p>Turning on <code class="docutils literal notranslate"><span class="pre">mechanics_printing()</span></code> makes the expressions significantly
shorter and is recommended. Alternatively, the <code class="docutils literal notranslate"><span class="pre">mprint</span></code> and <code class="docutils literal notranslate"><span class="pre">mpprint</span></code>
commands can be used.</p>
<p>If there are non-holonomic constraints, dependent speeds need to be specified
(and so do dependent coordinates, but they only come into play when linearizing
the system). The constraints need to be supplied in a list of expressions which
are equal to zero, trivial motion and configuration constraints are shown
below:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s1">&#39;N&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s1">&#39;q1 q2 q3 q4&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span><span class="p">,</span> <span class="n">q3d</span><span class="p">,</span> <span class="n">q4d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s1">&#39;q1 q2 q3 q4&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">,</span> <span class="n">u4</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s1">&#39;u1 u2 u3 u4&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1">#Here we will assume q2 is dependent, and u2 and u3 are dependent</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">speed_cons</span> <span class="o">=</span> <span class="p">[</span><span class="n">u2</span> <span class="o">-</span> <span class="n">u1</span><span class="p">,</span> <span class="n">u3</span> <span class="o">-</span> <span class="n">u1</span> <span class="o">-</span> <span class="n">u4</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">coord_cons</span> <span class="o">=</span> <span class="p">[</span><span class="n">q2</span> <span class="o">-</span> <span class="n">q1</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">q_ind</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">q_dep</span> <span class="o">=</span> <span class="p">[</span><span class="n">q2</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">u_ind</span> <span class="o">=</span> <span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u4</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">u_dep</span> <span class="o">=</span> <span class="p">[</span><span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">kd</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">u1</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">u2</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="n">u3</span><span class="p">,</span> <span class="n">q4d</span> <span class="o">-</span> <span class="n">u4</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">q_ind</span><span class="p">,</span> <span class="n">u_ind</span><span class="p">,</span> <span class="n">kd</span><span class="p">,</span>
<span class="gp">... </span>          <span class="n">q_dependent</span><span class="o">=</span><span class="n">q_dep</span><span class="p">,</span>
<span class="gp">... </span>          <span class="n">configuration_constraints</span><span class="o">=</span><span class="n">coord_cons</span><span class="p">,</span>
<span class="gp">... </span>          <span class="n">u_dependent</span><span class="o">=</span><span class="n">u_dep</span><span class="p">,</span>
<span class="gp">... </span>          <span class="n">velocity_constraints</span><span class="o">=</span><span class="n">speed_cons</span><span class="p">)</span>
</pre></div>
</div>
<p>A dictionary returning the solved <span class="math notranslate nohighlight">\(\dot{q}\)</span>’s can also be solved for:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">(</span><span class="n">pretty_print</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">kindiffdict</span><span class="p">()</span>
<span class="go">{q1&#39;: u1, q2&#39;: u2, q3&#39;: u3, q4&#39;: u4}</span>
</pre></div>
</div>
<p>The final step in forming the equations of motion is supplying a list of
bodies and particles, and a list of 2-tuples of the form <code class="docutils literal notranslate"><span class="pre">(Point,</span> <span class="pre">Vector)</span></code>
or <code class="docutils literal notranslate"><span class="pre">(ReferenceFrame,</span> <span class="pre">Vector)</span></code> to represent applied forces and torques.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s1">&#39;N&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">u</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s1">&#39;q u&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">qd</span><span class="p">,</span> <span class="n">ud</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s1">&#39;q u&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s1">&#39;P&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s1">&#39;Pa&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">BL</span> <span class="o">=</span> <span class="p">[</span><span class="n">Pa</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">FL</span> <span class="o">=</span> <span class="p">[(</span><span class="n">P</span><span class="p">,</span> <span class="mi">7</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="p">[</span><span class="n">q</span><span class="p">],</span> <span class="p">[</span><span class="n">u</span><span class="p">],</span> <span class="p">[</span><span class="n">qd</span> <span class="o">-</span> <span class="n">u</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">fr</span><span class="p">,</span> <span class="n">frstar</span><span class="p">)</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">kanes_equations</span><span class="p">(</span><span class="n">BL</span><span class="p">,</span> <span class="n">FL</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">mass_matrix</span>
<span class="go">Matrix([[5]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">forcing</span>
<span class="go">Matrix([[7]])</span>
</pre></div>
</div>
<p>When there are motion constraints, the mass matrix is augmented by the
<span class="math notranslate nohighlight">\(k_{dnh}(q, t)\)</span> matrix, and the forcing vector by the <span class="math notranslate nohighlight">\(f_{dnh}(q,
\dot{q}, u, t)\)</span> vector.</p>
<p>There are also the “full” mass matrix and “full” forcing vector terms, these
include the kinematic differential equations; the mass matrix is of size (n +
o) x (n + o), or square and the size of all coordinates and speeds.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">mass_matrix_full</span>
<span class="go">Matrix([</span>
<span class="go">[1, 0],</span>
<span class="go">[0, 5]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">forcing_full</span>
<span class="go">Matrix([</span>
<span class="go">[u],</span>
<span class="go">[7]])</span>
</pre></div>
</div>
<p>Exploration of the provided examples is encouraged in order to gain more
understanding of the <code class="docutils literal notranslate"><span class="pre">KanesMethod</span></code> object.</p>
</section>
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